Preface ....................................................... vii
Introduction .................................................... 1
1 First-Order Deformations ..................................... 5
1 The Hilbert Scheme ........................................ 5
2 Structures over the Dual Numbers .......................... 9
3 The Tl Functors .......................................... 18
4 The Infinitesimal Lifting Property ....................... 26
5 Deformations of Rings .................................... 35
2 Higher-Order Deformations ................................... 45
6 Subschemes and Invertible Sheaves ........................ 45
7 Vector Bundles and Coherent Sheaves ...................... 53
8 Cohen-Macaulay in Codimension Two ........................ 58
9 Complete Intersections and Gorenstein in Codimension
Three .................................................... 73
10 Obstructions to Deformations of Schemes .................. 78
11 Obstruction Theory for a Local Ring ...................... 85
12 Dimensions of Families of Space Curves ................... 88
13 A Nonreduced Component of the Hilbert Scheme ............. 91
3 Formal Moduli ............................................... 99
14 Plane Curve Singularities ............................... 100
15 Functors of Artin Rings ................................. 106
16 Schlessinger's Criterion ................................ 111
17 Hilb and Pic are Pro-representable ...................... 118
18 Miniversal and Universal Deformations of Schemes ........ 120
19 Versal Families of Sheaves .............................. 128
20 Comparison of Embedded and Abstract Deformations ........ 131
21 Algebraization of Formal Moduli ......................... 138
22 Lifting from Characteristic p to Characteristic 0 ....... 144
4 Global Questions ........................................... 149
23 Introduction to Moduli Questions ........................ 150
24 Some Representable Functors ............................. 156
25 Curves of Genus Zero .................................... 164
26 Moduli of Elliptic Curves ............................... 167
27 Moduli of Curves ........................................ 177
28 Moduli of Vector Bundles ................................ 188
29 Smoothing Singularities ................................. 199
References ................................................. 217
Index ......................................................... 225
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